The equivalence defined above is not robust in the sense that two states whose probability distributions differ very slightly are not equivalent.
Consider the PDFA \(A_\varepsilon \)  (c) in Fig. REF , where \(\varepsilon \in (0,0.5]\) . It maps every string \(a^n\) , \(n \ge 1\) , to \((0.5-\varepsilon )^{n-1} (0.5+\varepsilon )\) . This PDFA is not equivalent to PDFA \(A\)  in Fig. REF .
However, \(f_{A_\varepsilon }\)  tends to \(f_A\)  as \(\varepsilon \)  tends to 0. Moreover, \(\pi (q^{\prime }_1)\)  tends to \(\pi (q_1)\) .
Previous works have addressed this issue by introducing a tolerance parameter \(t\) .
In [1]}, states are compared using the probability of the last symbol of a non-empty string defined as:
\(\pi ^\ell _{q}(s\sigma ) = \pi ^\ast (s|q)(\sigma )&\;\mathrm {where}\; \sigma \in \Sigma _\$, s\in \Sigma ^\ast , q\in Q\) 
